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Elliot Anshelevich studies algorithms for networks with decentralized or strategic decision makers. His institutional research description includes network formation, algorithmic game theory, approximation algorithms and computational questions about collective choice.^[ref]

He is a coauthor of The Price of Stability for Network Design with Fair Cost Allocation by Elliot Anshelevich and others. The paper asks how good the best stable network can be when each user can individually change routes and shared edge costs are divided among their users. This separates the best equilibrium's cost from the worst equilibrium's cost, formalized by price of stability and price of anarchy.^[ref]

The question is useful for a designer who can recommend a network but cannot prevent unilateral departures. A low-cost recommendation is sustainable only if no user can improve by leaving it; computing a cheap network and understanding its incentives are therefore different parts of the problem.

For a cost-minimization game, the price of anarchy is the ratio of the worst equilibrium social cost to the best feasible social cost:Here $\mathcal{E}$ is the specified equilibrium set, $\mathcal{S}$ the feasible profiles, and $C$ the chosen nonnegative social cost. The ratio requires a positive denominator; zero optimum costs and empty equilibrium sets need separate conventions. A payoff-maximization formulation uses the reciprocal orientation to keep a larger number meaning worse efficiency.

In the two-edge routing game with one unit of flow and latencies $x$ and one, the equilibrium costs one while the optimum costs $3/4$, so the price of anarchy is $4/3$. The best equilibrium instead defines the price of stability. Bounds depend on the equilibrium notion and the allowed game class; one example does not establish a universal bound for all games.

For a cost-minimization game with positive social optimum and a nonempty set $\mathcal{E}$ of the specified Nash equilibria, the price of stability isIt measures the cost of requiring stability even when the best equilibrium can be selected. The equilibrium class, such as pure or mixed, and social-cost convention must be stated.

In a symmetric two-player game with actions $A,B$, let each player's cost be one at $(A,A)$, two at $(B,B)$ and three when the actions differ. Both coordinated choices are Nash equilibria; the best equilibrium has total cost two, matching the optimum, so the price of stability is one. The worst equilibrium instead costs four, giving price of anarchy two. The distinction matters when a designer can recommend an outcome but cannot force anyone to remain at an unstable one.^[ref]